Require
Export
Setoid_replace.
Section
Setoid.
Variable
A : Type.
Variable
Aeq : A -> A -> Prop.
Record
Setoid_Theory : Prop :=
{ Seq_refl : (x:A) (Aeq x x);
Seq_sym : (x,y:A) (Aeq x y) -> (Aeq y x);
Seq_trans : (x,y,z:A) (Aeq x y) -> (Aeq y z) -> (Aeq x z)
}.
End
Setoid.
Definition
Prop_S : (Setoid_Theory Prop iff).
Split; [Exact iff_refl | Exact iff_sym | Exact iff_trans].
Save
.
Hint
prop_set : setoid := Resolve (Seq_refl Prop iff Prop_S).
Hint
prop_set : setoid := Resolve (Seq_sym Prop iff Prop_S).
Hint
prop_set : setoid := Resolve (Seq_trans Prop iff Prop_S).
Add
Morphism or : or_ext.
Intros.
Inversion H1.
Left.
Inversion H.
Apply (H3 H2).
Right.
Inversion H0.
Apply (H3 H2).
Save
.
Add
Morphism and : and_ext.
Intros.
Inversion H1.
Split.
Inversion H.
Apply (H4 H2).
Inversion H0.
Apply (H4 H3).
Save
.
Add
Morphism not : not_ext.
Red ; Intros.
Apply H0.
Inversion H.
Apply (H3 H1).
Save
.
Definition
fleche [A,B:Prop] := A -> B.
Add
Morphism fleche : fleche_ext.
Unfold fleche.
Intros.
Inversion H0.
Inversion H.
Apply (H3 (H1 (H6 H2))).
Save
.